A Graph Matching Approach to Symmetry Detection and Analysis - Computational Intelligence Paradigms in Advanced Pattern Classification - page 140

times on an Intel(R) Core(TM)2 Duo running at 2.20GHz with 4GB of memory
are below 1 second, for creating the local descriptors, and less than 2 seconds for
computing the eigendecomposition using Matlab's built-in eigs function.

5.5 Additional Results

Figure 16 presents more results on synthetic images. Figures 17 and 18 show more
symmetry detection results on real images. The SSA successfully handles cases of
real images where symmetric objects are embedded in a cluttered background, have
few local features in their model representation, or underwent projective distortion.
If there are multiple symmetries in a single image, they can belong to a single
or to multiple objects as in Fig. 19. We present the self-alignments corresponding
to nine leading eigenvalues to demonstrate the SSA's ability to discover multiple
symmetric objects in an image.
Another example of 3D symmetry analysis is given in Fig. 20, where we show
the detected symmetry axes of a pyramid. Figure 20a depicts the eigenvalues of the
spectral self-alignment. We then applied the assignment pruning measure, which
allows us to detect the true assignments, corresponding to rotations and reflections.
We were able to detect all reflectional and rotational symmetries, but one.

6 Conclusions

We presented a computational scheme for the detection of symmetric structures
in n dimensional spaces. Our approach is based on the properties of the spectral

relaxation of the self-alignment problem. We showed that both reflectional and rota-
tional symmetries can be efficiently recovered by analyzing the spectral properties
of this formulation. Geometrical constraints were incorporated to improve the ro-
bustness of our scheme. We applied this approach to the analysis of image, by using
local features representations. The resulting scheme was shown experimentally to
be both efficient and robust as we are able to detect symmetric structure embedded
in clutter in real images.

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Appendix A

Theorem 3. Given two distinct reflectional transforms T D 1 and T D 2 , one can recover

the corresponding symmetrical rotational transforms T C K

T C K = T D 1 · T D 2 (17)

Proof. Given a set S with two reflectional symmetry axes. The center of the corre-
sponding rotational symmetry is at ( x 0 , y 0 ) and the axis of one of the reflection axes

is at an angle of α o . Then, a reflection transform T D K ,isgivenby